nLab Lagrangian correspondences and category-valued TFT

This entry describes classes of examples of A-∞ category-valued FQFTs defined on a version of the symplectic category.

Overview

Let $(X.\omega)$ be a compact symplectic manifold. At least in good cases to this is associated a Fukaya category $Fuk(X)$ of Lagrangian submanifolds and an enlarged version $Fuk^#(X)$.

Write $X^-$ for the symplectiv manifold $(X,-\omega)$.

Now if $(X_j, \omega_j)$ for $j = 0,1$ are two Lagrangian submanifolds and $L_{01} \subset X^-_0 \times X_1$ a Lagrangian correspondence then we get an A-∞ functor $\phi(L_{01}) : Fuk^#(X_0) \to Fuk^#(X_1)$

Theorem

(Wehrheim, Woodward)

For $L_{01} \subset X^{-}_0 \times X_1$ and $L_{12} \subset X^{-}_1 \times X_2$ Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an $A_\infty$-homotopy

$$\Phi(L_{01}) \circ \Phi(L_{12}) \simeq \Phi(L_{01} \circ L_{12}) \,,$$

where on the right we have a natural notion of composition of Lagrangian submanifolds.

(Wehrheim-Woodward 07, section 3.2)

This is the symplectic version of Mukai functors?.

Example

For $X_0$ and $X_1$ a compact Riemann surfaces and $M(X_0), M(X_1)$

their moduli spaces of fixed determinant rank $n$-bundles, and for $Y_{01}$ a cobordism (compact, oriented) from $X_0$ to $X_1$ then consider

$$L(Y_{01}) := Image( M(Y_{01}) \stackrel{restriction}{\to} M(X_0)^- \times M(X_1) )$$

If $Y_{01}$ is elementary in that there exists a Morse function $Y \to \mathbb{R}$ with $\leq 1$ critical points then $L(Y_{01})$ is a Lagrangian correspondence.

Corollary

The assignment

$$Y_{01} \mapsto \Phi(L(Y_{01}))$$

defines a 2+1-dimensional FQFT for connected cobordisms with values in A-∞ categories.

This is supposed to be the 2+1-dimensional part of Donaldson theory.

Other theories that fit into this framework:

1. symplectic Khovanov theory? (Seidel-Smith and Rezazodegab)

2. Heegard-Floer theory?

   $X$ a surface $\mapsto$ $Fuk^# sym X$

   elementary cobordisms $\mapsto$ $\Phi($vanishing cycle$)$

Lagrangian correspondences

Write $X^-j = (X_j , -\omega_j)$ for a symplectic manifold with its symplectic form reversed.

Definition

For $(X_j, \omega_j)$ two symplectic manifolds, a Lagrangian correspondence is a Lagrangian submanifold of $X^-_0 \times X_1$, that is

$$\iota : L_{0,1} \hookrightarrow X^-_0 \times X_1$$

with $dim(L_{0,1}) = \frac{1}{2}(dim(x_0) + dim(X_1))$

and

$$\iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0 \,,$$

where $\pi_i$ are the two projections out of the product.

The composition of two Lagrangian submanifolds is

$$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})$$

which is a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.

Example

For $\phi : X_0 \to X_1$ a symplectomorphism we have

$graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

Example

Let $X$ be a manifold, $G= U(n)$ the unitary group, $P \to X$ a $G$-principal bundle and $D \to X$ a $U(1)$-bundle with connection.

Then there is the moduli space $M(X) = M(P,D)$ of connections on $P$ with central curvature and given determinant.

For example if $X$ has genus $g$ then

$$M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}$$

such that $\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G$

Let $Y_{01}$ be a cobordism from $X_0$ to $X_1$ with extension

$$L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to} M(X_0)^- \times M(X_1) )$$

is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that

$$L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12}) \,.$$

References

• Katrin Wehrheim, Chris Woodward, Functoriality for Lagrangian correspondences in Floer theory (arXiv:0708.2851)

• Katrin Wehrheim, Chris Woodward, Floer Cohomology and Geometric Composition of Lagrangian Correspondences (arXiv:0905.1368)